Fractional Order Derivative Model of Viscoelastic layer for Active Damping of Geometrically Nonlinear Vibrations of Smart Composite Plates

Fractional Order Derivative Model of Viscoelastic layer for Active Damping of Geometrically Nonlinear Vibrationsof Smart Composite Plates Priyankar Datta1, Manas C.. Ray1 Abstract: This

Fractional Order Derivative Model of Viscoelastic layer for Active Damping of Geometrically Nonlinear Vibrations

of Smart Composite Plates

Priyankar Datta1, Manas C Ray1

Abstract: This paper deals with the implementation of the one dimensional form

of the fractional order derivative constitutive relation for three dimensional analysis

of active constrained layer damping (ACLD) of geometrically nonlinear laminatedcomposite plates The constraining layer of the ACLD treatment is composed ofthe vertically/obliquely reinforced 1–3 piezoelectric composites (PZCs) The vonKármán type nonlinear strain displacement relations are used to account for the ge-ometric nonlinearity of the plates A nonlinear smart finite element model (FEM)has been developed Thin laminated substrate composite plates with various bound-ary conditions and stacking sequences are analyzed to verify the effectiveness ofthe three-dimensional FDM for both the passive and active control authority of theACLD patch located at the center of the laminates

Keywords: Fractional Derivative; Smart Structures; Nonlinear Vibration; ActiveControl

1 Introduction

Controlling vibration levels in structures under dynamic loading is an importantaspect in the study of dynamics and control as high vibration often generates noiseand can lead to cyclic fatigue failure of the structure The drive to use lightweightcomposite structures, particularly in aerospace, automotive and marine industries,makes them more vibration prone The composite materials provide high strength

to weight ratio and enhanced performance but most of them have very little inherentdamping to offer Thus the incorporation of damping into the flexible host structure

by bonding a viscoelastic layer to the same became one of the popular approaches

to reduce the noise and vibration of host structure and the viscoelastic layer iscalled as the free-layer or ‘unconstrained’ treatment [Stanway, Rongong, and Sims(2003)]

1 Mechanical Engineering Department, Indian Institute of Technology, Kharagpur 721302, India

The direct and the converse piezoelectric effects inherently present in the tric materials enabled them to act as distributed actuators and sensors, respectively.Instead of attaching the piezoelectric layer directly to the host structure, if it isbonded to the structure using a viscoelastic layer such that the viscoelastic layer isconstrained between the host structure and the piezoelectric layer, the active con-strained layer damping (ACLD) treatment is formed [Baz and Ro (1995)] Theflexural vibration control by the constrained layer damping treatment is due to thedissipation of energy in the constrained viscoelastic layer As the dissipation ofenergy is mostly due to the transverse shear deformation of the constrained layer,

piezoelec-it can be increased by improving the transverse shear deformations of the same.When activated by suitable control voltage, the constraining piezoelectric layer ofthe active constrained layer damping (ACLD) treatment increases the transverse s-hear deformation of the constrained viscoelastic layer over its passive counterpartresulting in active or smart damping of the host structure If the piezoelectric layer

is left inactive, the standard passive constrained layer damping (PCLD) is achieved.The use of piezoelectric materials has been extensively investigated by several re-searchers [Ha, Keilers, and Chang (1992); Hwang and Park (1993); Baz and Ro(1994); Shen (1994); Baz and Ro (1995); Yi, Ling, and Ying (1998); Ray (1998);

Yi and Sze (2000); Balamurugan and Narayanan (2002); Ray and Mallik (2005);Ray (2007); Ray and Shivakumar (2009); Ray, Dong, and Atluri (2015)] for activecontrol of flexible lightweight structures

Piezoelectric composite (PZC) materials are the new class of distributed smart terials in which piezoelectric fiber reinforcements are used along with epoxy asmatrix Among the different types of PZC materials studied by the researchers,the vertically and the obliquely reinforced 1–3 PZC materials are commerciallyavailable [Gentilman, Fiore, Houston, and Corsaro (1999)] Recently, Ray, andhis co-researchers [Ray and Pradhan (2006); Ray and Pradhan (2007); Ray andBatra (2007); Panda and Ray (2009a); Biswas and Ray (2013); Kundalwal, Ku-mar, and Ray (2013); Kanasogi and Ray (2013)] analyzed the performance of these1–3 PZC materials for active damping of linear and geometrically nonlinear vibra-tions of composite and functionally graded beams, plates and shells For the timedomain analysis, the constrained viscoelastic layer of the ACLD treatment is mod-eled by the Golla-Hughes-McTavish (GHM) [Golla and Hughes (1985); McTavishand Hughes (1993)] method Golla and Hughes (1985) proposed a time domainmodel with ordinary integer differential operators where hereditary integral for-

ma-m of the viscoelastic constitutive law is used to forma-mulate a finite elema-ment ma-model(FEM) Later [McTavish and Hughes (1993)] extended the Golla-Hughes modeland formulated the Golla-Hughes-McTavish (GHM) model representing the mate-rial modulus as a series of mini-oscillators Using GHM model the FEM is first

derived in the Laplace domain and the resulting FEM is retransformed to obtain theFEM in time-domain Auxiliary dissipative coordinates are used to model the en-ergy dissipation of the viscoelastic material Several other methods for modellinglinear viscoelastic material are available in the open literature [Lesieutre and Min-gori (1990); Lesieutre and Bianchini (1995); Bagley and Torvik (1983a); Bagleyand Torvik (1983b)] Lesieutre and Mingori (1990) developed a frequency depen-dent linear viscoelastic model in terms of augmenting thermodynamic fields (ATF)that are coupled with the mechanical displacement field The ATF model intro-duces additional thermal coordinates to account for the dissipation Lesieutre andBianchini (1995) introduced another time-domain model of linear viscoelasticitybased on the anelastic displacement field (ADF) in three-dimensional form In thismodel, the dissipation is modeled by considering anelastic component of the dis-placement field in addition to the elastic counterpart Bagley and Torvik (1983a)established a link between the molecular theories to predict the macroscopic behav-ior of certain viscoelastic media and an empirically developed fractional calculusapproach to viscoelasticity Later using the fractional calculus approach they pro-posed an effective time domain model of the viscoelastic material named fractionalderivative model (FDM) [Bagley and Torvik (1983b); Bagley and Torvik (1985)].Modelling of viscoelastic material in time domain for implementing different mod-ern control strategies is an important issue particularly in case of active damping ofnonlinear vibrations of three-dimensional structures The GHM method has beenextensively used to model the constrained viscoelastic layer in time-domain [Pandaand Ray (2012); Panda and Ray (2009b); Sarangi and Ray (2010); Kumar and Ray(2013); Kattimani and Ray (2014a); Kattimani and Ray (2014b); Kanasogi andRay (2015)] for active damping of linear and geometrically nonlinear vibrations ofbeams, plates and shells However, as already mentioned the GHM method need-

s to introduce additional dissipative coordinates resulting in the twofold increase

in the overall generalized degrees of freedom of the overall structure Hence, thecomputational cost is also enormously increased limiting the use of the GHM par-ticularly in case of active control of geometrically nonlinear vibrations The FDM

of viscoelastic material does not require any additional generalized dissipative ordinates and appears to be an efficient method for modelling viscoelastic layer intime-domain Galucio, Deü, and Ohayon (2004) presented a finite element modelfor transient dynamic analysis and ACLD of sandwich beams [Galucio, Deü, andOhayon (2005)] using one-dimensional fractional derivative constitutive equationsfor viscoelastic layer However, the study on the effectiveness of this fractionalderivative model for three-dimensional analysis of both PCLD and ACLD of ge-ometrically nonlinear vibrations of composite structures is not yet available in theopen literature

co-The present work is concerned with the implementation of the one-dimensionalfractional order derivative constitutive relation of viscoelastic material for three-dimensional analysis of passive and active constrained layer damping of geomet-rically nonlinear vibrations of smart composite plates Needless to say that thefinite element method has been established as an efficient tool for numerical analy-sis of structures [Dong, El-Gizawy, Juhany, and Atluri (2014a); Dong, El-Gizawy,Juhany, and Atluri (2014b)] Hence, a three dimensional finite element model hasbeen developed In this model von Kármán strain-displacement relations are con-sidered for incorporating the geometric nonlinearity Third order shear deformationtheory (TSDT) [Reddy (2004)] or zeroth-order shear deformation theory (ZSDT)[Ray (2003); Datta and Ray (2015)] provides accurate analysis for thin and thickplates but if they are used for the host plates, transverse shear stresses at the top andbottom surfaces of the plates will be zero But at the interface between the patch

of the ACLD treatment and the substrate plate, transverse shear stresses cannot bezero and must be modeled Thus TSDT or ZSDT cannot be used for modeling s-mart structure with patch type piezoelectric actuator On the other hand, althoughhigher order shear deformation theory (HSDT) presented by Lo, Christensen, and

Wu (1977) can be used for accurate analysis of smart structures, the computationalcost of the analysis highly increases because of large number of generalized de-grees of freedom For the present analyses thin plates have been considered andconsequently the axial displacement fields based on the first order shear deforma-tion theory (FSDT) is used to model the axial displacements in all the layers ofthe overall plate Substrate laminated plates with various stacking sequences andboundary conditions have been presented to check the efficacy of the passive andactive control authority of the ACLD patch using the derived finite element model

Figure 1: Schematic representation of a laminated composite plate integrated withthe patch of ACLD treatment composed of constrained viscoelastic layer and 1–3PZC constraining layer

2 Basic equations

Figure 1 illustrates a rectangular substrate laminated plate integrated with a gular patch of the ACLD treatment at the top surface of the plate The constraininglayer of the patch of the ACLD treatment is made of the vertically/obliquely rein-forced 1-3 PZC material with a thickness denoted by hpwhile the thickness of thelaminated substrate plate and the constrained viscoelastic layer is represented by

rectan-hand hv, respectively The substrate plate is composed of N number of tional fiber-reinforced orthotropic layers The laminate co-ordinate system (xyz) ischosen in such a way that the mid-plane of the substrate laminate represents thereference plane while the lines x = 0, a and y = 0, b represent the boundaries ofthe same The thickness co-ordinates zof the top and the bottom surfaces of any kth(k = 1, 2, 3 N + 2) layer of the overall plate are represented by hk+1 and hk, re-spectively θkis the fiber orientation in any layer of the substrate plate in the plane(xy) of the lamina with respect to laminate co-ordinate system As mentioned ear-lier, the reinforcing piezoelectric fibers in the constraining layer of the patch may

unidirec-be vertical (Fig 2(a)) or coplanar with the vertical xz (Fig 2(b)) or yz (Fig 2(c))

(c)Figure 2: (a) Lamina of vertically reinforced 1-3 PZC; (b) Lamina of 1–3 PZCwhere the piezoelectric fibers are coplanar with the vertical xz plane.; (c) Lamina

of 1–3 PZC where the piezoelectric fibers are coplanar with the vertical yz plane

(a) (b)Figure 3: Kinematics of deformation of the plate in the xz and yz planes.

plane making an angle ψ with respect to z axis The kinematics of axial mations of the overall plate based on the layerwise FSDT has been illustrated inFig 3 Displayed in this figure, the variables u0 and v0 are the generalized trans-lational displacements of a point (x, y) on the reference plane (z = 0) along x- andy-directions, respectively θx, φx and γxdenote the generalized rotations of the nor-mals to the middle planes of the substrate plate, the viscoelastic layer and the PZClayer, respectively in the xz plane while θy, φy and γy represent their generalizedrotations in the yz plane According to the kinematics of deformation illustrated inFig 3, the axial displacements u(x, y, z,t) and v(x, y, z,t) of a point in any layer ofthe overall plate along the x and y directions, respectively, can be expressed as

defor-u(x, y, z,t) = u0(x, y,t) + (z − hz − h/2i) θx(x, y,t)

+ (hz − h/2i − hz − hN+2i) φx(x, y,t) + hz − hN+2i γx(x, y,t) andv(x, y, z,t) = v0(x, y,t) + (z − hz − h/2i) θy(x, y,t)

+ (hz − h/2i − hz − hN+2i) φy(x, y,t) + hz − hN+2i γy(x, y,t) (1)

where, a function within the bracket h i represents the appropriate singularity tions such that the interface continuity of displacements between the substrate andthe viscoelastic layer and between the viscoelastic layer and the constraining layerare satisfied Since the objective of the present analysis is to control the flexuralvibrations of the substrate plate and the magnitude of the transverse piezoelectriccoefficient of the 1-3 PZC layer is much greater than that of the in-plane piezoelec-tric coefficients of the same, the transverse normal strain in the overall plate must

func-be considered in the model The transverse displacement w(x, y, z,t) at any point inthe overall structure may be assumed to have quadratic variation with respect to z

across their thicknesses and can be expressed as

w(x, y, z,t) = w0(x, y,t) + zθz(x, y,t) + z2φz(x, y,t) (2)

in which w0 refers to the transverse displacement at any point on the referenceplane, θz and φz are the generalized displacements representing the gradient andthe second order derivative of the transverse displacement in the overall structurewith respect to the thickness coordinate (z) Such quadratic variation of transversedisplacement also provides parabolic distribution of transverse shear stress acrossthe thickness of the overall plate For introducing geometric nonlinearity in themodel, von Kármán type strain displacement relations are considered as follows:

where εx, εy, εzare the normal strains along x, y and z directions, respectively, γxy

is the in-plane shear strain and γxz, γyz are the transverse shear strains By usingthe displacement fields (Eqs (1) and (2)) and the nonlinear strain displacementrelations (Eq (3)), the bending and shear strain vectors are obtained The bendingstrain vectors {εb}c, {εb}v and {εb}p in the substrate composite plate, the con-strained viscoelastic layer and the active constraining layer, respectively, can beexpressed as

{εb}c= {εbt} + [Z1] {εbr} + {εbnl} ; {εb}v= {εbt} + [Z3] {εbr} + {εbnl} and

Similarly, the transverse shear strain vectors {εs}c, {εs}vand {εs}pin the substratecomposite plate, the constrained viscoelastic layer and the active constraining layer,respectively, are given by:

{εs} = {εst} + [Z2] {εsr} ; {εs} = {εst} + [Z4] {εsr} and

{εs}p= {εst} + [Z6] {εsr} (7)

The matricesZβ (β = 1, 2, , 6) appearing in Eqs (6) and (7) have been defined

in the Appendix, while the generalized strain vectors are given by

{σb} = [σx σy σxy σz]T and {σs} = [σxz σyz]T (9)

where σx, σy, σzare the normal stresses along x, y and z directions, respectively,

σxyis the in-plane shear stress and σxz, σyzare the transverse shear stresses

2.1 Constitutive relation for the orthotropic material

As the laminated composite plate is made of several orthotropic layers with theirprincipal material axes arbitrarily oriented with respect to the laminate coordinatesystem (xyz), the constitutive equations of each layer is transformed to the laminatecoordinate system Thus the stress–strain relations for the kth off-axis lamina withrespect to the laminate coordinate system can be arranged in the context of thepresent formulation as follows:

σsk

o

=h

2.2 Fractional order constitutive modeling of the viscoelastic material

The three-dimensional fractional order constitutive relations for the viscoelasticlayer is derived using the four-parameter one-dimensional fractional order deriva-tive constitutive relation The one-dimensional fractional order derivative constitu-tive equation introduced by Bagley and Torvik (1983a) is given by

as [Schmidt and Gaul (2002)]:

and similarly the transverse shear strainsγv

For an isotropic material, the three-dimensional normal strains εxv, εyv and εzv along

x, y and z directions, respectively under three-dimensional state of stresses are givenby

while the elastic coefficients Ci jv are presented in the Appendix

At this juncture two anelastic strain functions can be introduced as follows:

{σv

b} =E∞

E0 [Cv] ({εbv} − {¯εv

b}) ;{σv

s} = G∞({εsv} − {¯εv

where the anelastic bending strain function ¯εv

b and the anelastic transverse shearstrain function { ¯εsv} are given by

{ ¯εbv}n+1= (1 − c)E∞− E0

E∞ {εbv}n+1− c

Nt

∑j=1

Aj+1{ ¯εv

where c is a dimensionless parameter given by c = τα

τ α +(∆t) α.Finally, substituting the relations given in Eq (22) into Eq (19), the discretizedforms of the constitutive relations of the viscoelastic material are obtained as fol-lows:

whereCv

b,  ¯Cbv, [Cv] andC¯v are explicitly given in the Appendix

2.3 Constitutive relation for the 1-3 piezo-composite (PZC) material

The constraining 1–3 PZC layer is considered to be subjected to the electric field

Ez along the z-direction only Accordingly, the constitutive relations for the straining layer of the ACLD treatment are given by [Ray and Pradhan (2006)]

the vertical xz- or yz-plane and the corresponding elastic coupling constant matrix

according as the piezoelectric fibers are coplanar with the vertical xz- or yz-plane

If the fibers of the PZC are coplanar with both the xz- and the yz-planes i.e

vertical-ly reinforced, this coupling matrix becomes a null matrix Also, the piezoelectricconstant matrices {eb} and {es} appearing in Eq (24) contain the following trans-formed effective piezoelectric coefficients of the 1–3 PZC:

{eb} = [ ¯e31 e¯32 e¯36 e¯33]T and {es} = [ ¯e35 e¯34]T (26)

The principle of virtual work is employed to derive the governing equations of theoverall smart structure and can be expressed as

3 Finite element formulation

The overall plate is discretized by eight-noded isoparametric quadrilateral elements.Following Eq (4), the generalized displacement vectors, associated with the ith(i = 1, 2, 3, , 8) node of the element can be written as

{dt} = [Nt] {de} and {dr} = [Nr] {de} (29)

while It and Ir are the (3 × 3) and (8 × 8) identity matrices, respectively and ni

is the shape function of natural coordinates associated with the ith node Makinguse of the relations given by Eqs (6) to (8) and (29), the strain vectors at any pointwithin the element can be expressed in terms of the nodal generalized displacementvectors as follows:

{εb}c= [Btb] {dte} + [Z1] [Brb] {der} +1

2[B1] [B2] {d

e

t} ;{εb}v= [Btb] {dte} + [Z3] [Brb] {der} +1

2[B1] [B2] {d

e

t} ;{εb}p= [Btb] {dte} + [Z5] [Brb] {der} +1

2[B1] [B2] {d

e

t} ;{εs}c= [Bts] {dte} + [Z2] [Brs] {dre} ;

t and  ¯de are the generalized anelastic nodal displacement vectors andhave the similar forms of {dte} and {de

r}, respectively

On substitution of Eqs (31) and (33) into Eq (27) and recognizing that Ez=

−V /hpwith V being the applied voltage across the thickness of the piezoelectriclayer, one can derive the following open-loop equations of motion of an elementintegrated with the ACLD treatment:

rr] {dre} = − { ¯Frne} −Fe

The elemental mass matrix ([Me]), the elemental stiffness matrices ([Kttne ], [Ktrne ],[Krtne ] and [Krre]), the elemental memory load vectors due to the viscoelastic material({ ¯Ftne} and { ¯Frne}), the elemental electroelastic coupling vectors (Fe

t pn and Fe

r p )and the elemental load vector {Fe} appearing in Eq (34) are given by

Aj+1d¯e

r

n+1− j;[ ¯Kttne ] = [ ¯Ktbe] + [ ¯Ktse] + [ ¯Ktbne ] ; [ ¯Ktrne ] = [ ¯Ktrbe ] + [ ¯Ktrse ] + [ ¯Ktrbne ] ;

[ ¯Krtne ] = [ ¯Ktrbe ]T+ [ ¯Ktrse ]T+1

2[ ¯K

e trbn]T; [ ¯Krre] = [ ¯Krrbe ] + [ ¯Krrse ] ;

of the overall structure integrated with the ACLD patch at any (n + 1)th time step

3.1 Computation of memory load due to the effect of viscoelastic materialUsing the relation between the strain vectors and the anelastic strain vector, therelation between generalized global displacement vector and global anelastic dis-placement vector can be obtained as follows:

{ ¯X }n+1= (1 − c)E∞− E0

E∞ {X}n+1− c

N t

∑j=1

It may be mentioned here that the Grünwald coefficients (Aj) in Eq (39) decreaseswith the increase in the value of j Thus the response of the viscoelastic mate-rial at a particular time depends more strongly on the recent past history of theresponse than on the remote past history of the response Therefore the Grünwaldcoefficients corresponding to the large value of j describe the fading memory of theviscoelastic layer and truncation of the Grünwald series after some value of j doesnot affect the response Substituting Eq (39) in Eq (38), the open-loop globalequations of motion of the overall structure is obtained in time domain as follows:

[M]X¨

n+1+ [Ktn∗] {X }n+1+ [Ktrn∗ ] {Xr}n+1= {F}n+1+ {Ftn∗}n+1− {Ft pn}V ;[Krtn∗ ] {X }n+1+ [Krrn∗ ] {Xr}n+1= {Frn∗}n+1− {Fr p}V (40)

in which [Ktn∗], [Ktrn∗ ], [Krtn∗ ], [Krr∗], {Ftn∗}n+1and {Frn∗}n+1 are given by

[Ktrn∗ ] = [Ktrn] + 1 − c

c

  E∞− E0E

[ ¯Ktrn] ;